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Theorem abeq2f 3010
Description: Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.) Avoid ax-13 2381. (Revised by Wolf Lammen, 13-May-2023.)
Hypothesis
Ref Expression
abeq2f.0 𝑥𝐴
Assertion
Ref Expression
abeq2f (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))

Proof of Theorem abeq2f
StepHypRef Expression
1 abeq2f.0 . . 3 𝑥𝐴
2 nfab1 2976 . . 3 𝑥{𝑥𝜑}
31, 2cleqf 3007 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}))
4 abid 2800 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
54bibi2i 339 . . 3 ((𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ (𝑥𝐴𝜑))
65albii 1811 . 2 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑥𝜑}) ↔ ∀𝑥(𝑥𝐴𝜑))
73, 6bitri 276 1 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1526   = wceq 1528  wcel 2105  {cab 2796  wnfc 2958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960
This theorem is referenced by:  rabid2f  3380  mptfnf  6476
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